I realize now that my concerns about the commonly accepted story of presocratic philosophy have led me to share Patricia Curd's view of its shortcomings, although I don't think I can accept her own alternative story.
Here is the problem. There is a 'Standard Account' (as Curd calls it) of the history of presocratic philosophy, which in outline is embraced by the most common texts and books on the subject in English: Barnes' two volume treatment; his popular and concise Penguin volume; KRS; Guthrie; McKirahan; and Hussey. This account is heavily influenced by Owen's "Eleatic Questions".
But the Standard Account is incoherent, because it puts forward an inconsistent triad:
1. Parmenides was influential.
2. Parmenides used arguments in philosophical logic to conclude that reality is single, unchanging, and perfect.
3. Parmenides' successors did not engage in philosophical logic and held that reality is plural, changing, and imperfect.
As Curd remarks:
On this standard account, the later Presocratics, such as the Pluralists and the Atomists, failing to see the futility of engaging in cosmological inquiry, try to answer Parmenides by agreeing that coming-to-be and passing-away are not real. Nevertheless, they still attempt to give rational accounts of the changing sensible world, insisting that a plurality of unchanging things underlie and explain that world. Moreover, these later Presocratics give no attention to problems of reference and meaning, offering no evidence that they recognized these issues as Parmenides' concerns. So either they must have badly misunderstood Parmenides, or else their answers were quite feeble (Legacy of Parmenides, paperback version, p. 12).There seem to be only two ways out: either (i) deny that Parmenides was (in any non-superficial sense) influential for later presocratics, or (ii) hold that he did importantly influence them, but that his arguments and conclusions were not as the Standard View maintains.
On the standard interpretation we must either attribute serious misunderstandings of Parmenides to later thinkers or claim that they felt free to ignore his arguments while accepting certain of his assumptions about what-is (13).
Curd defends (ii). Parmenides, she says, did not assert 'numerical monism', that only one thing really exists. He wanted only to insist upon 'predicational monism' (as she calls it), viz. that whatever exists has to be (in a suitably strong sense) one. It is consistent with 'predicational monism', she asserts, that many sorts of things exist, and that these, although changeless in themselves, change relative to one another.
I rather favor (i), in a qualified form. I do so, briefly, because it seems to me that predicational monism, by a very quick route (one that Parmenides could not have but seen), implies numerical monism; also, because we can allow that Parmenides' influence was limited and 'negative'--in the sense that he points to a (relatively circumscribed) deficiency in the earlier tradition-- rather than being extensive and positive. (Compare the influence of Hume on Reid, versus his influence on Kant.) But this is for later posts.